Communications in Mathematical Physics

, Volume 108, Issue 3, pp 489–526 | Cite as

Sharpness of the phase transition in percolation models

  • Michael Aizenman
  • David J. Barsky


The equality of two critical points — the percolation thresholdp H and the pointp T where the cluster size distribution ceases to decay exponentially — is proven for all translation invariant independent percolation models on homogeneousd-dimensional lattices (d≧1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameterM(β,h), which forh=0 reduces to the percolation densityP — at the bond densityp=1−eβ in the single parameter case. These are: (1)MhM/∂h+M2MM/∂β, and (2) ∂M/∂β≦|J|MM/∂h. Inequality (1) is intriguing in that its derivation provides yet another hint of a “ϕ3 structure” in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents\(\hat \beta\) and δ. One of these resembles an Ising model inequality of Fröhlich and Sokal and yields the mean field bound δ≧2, and the other implies the result of Chayes and Chayes that\(\hat \beta \leqq 1\). An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation\(\hat \beta (\delta - 1) \geqq 1\) and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.


Neural Network Phase Transition Long Range Cluster Size Quantum Computing 
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  1. 1.
    Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys.74, 41–59 (1980)Google Scholar
  2. 2.
    Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 229–237 (1981)Google Scholar
  3. 3.
    Aizenman, M., Newman, C.M.: Discontinuity of the percolation density in one-dimensional 1/|x−y|2 percolation models. Commun. Math. Phys.107, 611–647 (1986)Google Scholar
  4. 4.
    Aizenman, M., Chayes, J.T., Chayes, L., Imbrie, J., Newman, C.M.: An intermediate phase with slow decay of correlations in one-dimensional 1/|x−y|2 Ising and Potts models (in preparation)Google Scholar
  5. 5.
    Aizenman, M.: Contribution in: Statistical physics and dynamical systems (Proceedings Kösheg 1984). Fritz, J., Jaffe, A., Szasz, D. (eds.). Boston: Birkhäuser 1985Google Scholar
  6. 6.
    Chayes, J.T., Chayes, L.: Critical points and intermediate phases on wedges of ℤd. J. Phys. A (to appear)Google Scholar
  7. 7.
    Hammersley, J.M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist.28, 790–795 (1957)Google Scholar
  8. 8.
    Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys.36, 107–143 (1984)Google Scholar
  9. 9.
    Newman, C.M., Schulman, L.S.: One-dimensional 1/|j − i|s percolation models: The existence of a transition fors≦2. Commun. Math. Phys.104, 547–571 (1986)Google Scholar
  10. 10.
    Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys.11, 790–795 (1970)Google Scholar
  11. 11.
    Newman, C.M.: Shock waves and mean field bounds. Concavity and analyticity of the magnetization at low temperature. Appendix to contribution in Proceedings of the SIAM workshop on multiphase flow, G. Papanicolau (ed.) (to appear)Google Scholar
  12. 12.
    Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys.44, 393–454 (1986)Google Scholar
  13. 13.
    Harris, A.B., Lubensky, T.C., Holcomb, W.K., Dasgupta, C.: Renormalization group approach to percolation problems. Phys. Rev. Lett.35, 327–330 (1975)Google Scholar
  14. 14.
    Chayes, J.T., Chayes, L.: An inequality for the infinite cluster density in Bernoulli percolation. Phys. Rev. Lett.56, 1619–1622 (1986)Google Scholar
  15. 15.
    Fröhlich, J., Sokal, A.D.: The random walk representation of classical spin systems and correlation inequalities. III. Nonzero magnetic field (in preparation)Google Scholar
  16. 16.
    Fernández, R., Fröhlich, J., Sokal, A.D.: Random-walk models and random-walk representations of classical lattice spin systems (in preparation)Google Scholar
  17. 17.
    Griffiths, R.B.: Correlations in Ising ferromagnets. II. External magnetic fields. J. Math. Phys.8, 484–489 (1967)Google Scholar
  18. 18.
    Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Submitted to Commun. Math. Phys.Google Scholar
  19. 19.
    van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab.22, 556–569 (1985)Google Scholar
  20. 20.
    Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982Google Scholar
  21. 21.
    Griffeath, D.: The basic contact process. Stochastic Processes Appl.11, 151–185 (1981)Google Scholar
  22. 22.
    Aizenman, M., Barsky, D.J., Fernández, R.: The phase transition in a general class of Ising-type models in sharp. Submitted to J. Stat. Phys.Google Scholar
  23. 23.
    Chayes, J.T., Chayes, L., Newman, C.M.: Bernoulli percolation above threshold: an invasion percolation analysis. Ann. Probab. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Michael Aizenman
    • 1
  • David J. Barsky
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

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